Naked Candidates

'Naked' in this context refers to all the remaining possible candidates on a cell which are going to be used in a strategy. The simplest such situation is a Naked Single - or the last remaining candidate on a cell. Generally speaking, if you are making notes on a Sudoku board you have reached a point where simple scanning of the rows, columns and boxes has brought you no further solutions. But you will be finding plenty of Singles on the easier puzzles, and hopefully not too few on the hardest ones.

A Naked Single is exactly equivalent to saying "Ah Ha! Looking at that cell I can see every other number either in the same box, the same row or the same columns, it's the only number that can fit"

Hidden candidates, mentioned below with regard to Pairs and so on, also have a Hidden Single Equivalent. It occurs when you find a cell with lots of possible but you reason "well, X can't go anywhere else in either the row, column or box, so it must go here.

Naked Pairs

A Naked Pair (also known as a Conjugate Pair) is a set of two candidate numbers sited in two cells that belong to at least one unit in common. That is they reside in the same row, box or column.

It is clear that the solution will contain those values in those two cells and all other candidates with those numbers can be removed from whatever unit(s) they have in common.

Naked Pairs examples
: Load Example
or : From the Start
In this example several Naked Pairs are available and I have highlighted two. In red in row A cells A2 and A3 both contain 1 and 6. We don't know which way round the 1 and the 6 will eventually be - we will find out later as we finish the puzzle, but it means we can remove all other 1s and 6s in the row - which the solver has highlighted in yellow. But A2 and A3 are also in the same box - so we can clear of the 1 in C1 as well.

The [6,7] in row C is also a Naked Pair. It is aligned just in the row but it removes three other candidates 6s and 7s in the row. Combining both Naked Pairs we get a solved cell of 8 in C1.

There are other Naked Pairs at this point. You can identify them yourself or load the puzzle up in the solver to see them,

Figure 2
: Load Example
or : From the StartJust to show that pairs don't have to be aligned on a row or column, in this group of pairs we have a [4,7] on H2 and J1 which remove some 7s in the same box. Two other Naked Pairs eliminate further candidates at this stage.

Naked Triples

A Naked Triple is slightly more complicated because it does not always imply three numbers each in three cells.

Any group of three cells in the same unit that contain IN TOTAL three candidates is a Naked Triple. Each cell can have two or three numbers, as long as in combination all three cells have only three numbers. When this happens, the three candidates can be removed from all other cells in the same unit.

The combinations of candidates for a Naked Triple will be one of the following:

The last case is interesting and the advanced strategy XY-Wings uses this formation.

Naked Triple
: Load Example
or : From the Start
This first example is as straightforward as it gets. In row E, centre box, is are the cells E4, E5 and E6 containing [5,8,9], [5,8] and [5,9] respectively. In total those three cells contain [5,8,9] so we have fixed those numbers in those cells - just not which way round they will be. This allows us to remove those numbers from the rest of the unit the Triple is aligned on - namely the row.

Naked Triples
: Load Example
or : From the Start
We have two Naked Triples at the same time on this board, in columns 1 and 9. There is no trickery in these Triples because the cells that form the triples are the last three unsolved cells in those columns - so they are bound to contain the three remaining values. Given that fact we can clear out those values from the box (and only the box since there is nothing to clear off in the columns). But the manoeuvre nets us a great deal of candidates and we get a solution of 9 in F8.

In terms of the candidates per cell the columns 1 triple is a {2/2/3} formation (reading down) and the second is {3/2/3}.

Naked Quads

A Naked Quad is rarer, especially in its full form but still useful if they can be spotted. The same logic from Naked Triples applies, but the reason it is so rare is because if a Quad is present the remaining cells are more like to be a Triple or Pair and the solver will highlight those first.

Naked Quad example
: Load Example
or : From the StartWell, I can't find an example in my 2012 stock, so I'm going to use the one found by Pieter from Australia. It's a cluster of cells in box 1. A1, B1, B2 and C1 collectively contain [1,5,6,8] so those numbers must occupy those cells. That allows us to remove the yellow highlighted candidates.

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Tuesday 15-Apr-2014

... by: Brian Fink

Regarding my previous comment, it doesn't matter how many cells are in the Naked Pair chain. There will be times when the pair itself will cancel out in the cell, because there is an odd number of overlapping Naked Pairs; but my example focuses on the quasi-complete Naked Pair vicious cycle, in which, if the cell in question was just that pair of digits, the puzzle would have 2 solutions. This trick eliminates that possibility from the equation.

Saturday 12-Apr-2014

... by: Brian Fink

I've discovered a version of Naked Pairs that is very helpful with just one cell (as opposed to an entire row/column/box).

Let's say we have a chain of an overlapping even number of Naked Pairs, all the same pair, with two ends that can see a single cell but are not both aligned on the same row or column with it or even sharing the same box. Then the pair can be removed from that cell and only that cell, leaving all other digits in that cell as possibilities.

For example, if we had C1=3/7/8, C7=7/8, E7=7/8, and E1=7/8, then in addition to removing 7/8 from Row C and Column 7, you may remove it from C1 as well.

There may also be a variation of this that only includes a cell that can be seen by two cells with naked pairs (same pair) that are not part of a traditional Naked Pair, but I have yet to prove that one.

Monday 7-Apr-2014

... by: Jan

In the Naked Quad section is stated "Well, I can't find an example in my 2012 stock, ..."

But you do have one! In the Naked Triple example is also a naked quad for 2, 3, 8 and 9 in D9, E9, F9 and F8 (and so also a Hidden Quad for 1, 4, 5 and 7 in D7, D8, E7 and F7)

Sunday 2-Mar-2014

... by: Pete

Ahh, I get it now.....Square C1 contains only numbers which are part of the 4 numbers in the quad and no extra numbers. Whereas square B3 contains extra numbers which are not part of the quad.

It only took me a month to figure out.....

Saturday 8-Feb-2014

... by: Pete

In the naked quad example, I don't get how square c1 is identified as part of the quad, when square b3 contains 3 of the 4 quad numbers. The only thing I notice is that b3 doesn't contain number 1.Is it the case that each of the quad squares has to contain one number which appears in all 4 squares? If so, then how does this tie-in with the naked triple possibility of (12) (23) (13) ? Here, there isn't a number which appears in all 3 cells..

Great site btw

Wednesday 22-Jan-2014

... by: KeithD

Quick suggestion for people who still can't understand the naked triple explanations, above and in the comments: try using one of the three numbers in the triple as the solution for any of the candidates for elimination (in the row, column or box as appropriate). Now, only two of the cells in the triple can be solved, while the third has no valid candidate. Clearly, only the three triple cells can contain the three triple numbers.

Eg, in the first triple example, try putting 5 in E1. Now E5 is 8 and E6 is 9, but E4 has no candidate.

Sunday 22-Sep-2013

... by: Tim

Naked Pairs figure 2 shows (1,2) at A6 and G6.Why not G4 and G6

Andrew Stuart writes:

That is a Naked Pair on G4/G6, but it doesn't lead to any eliminations (in the row) so it doesn't get highlighted

Sunday 30-Jun-2013

... by: Klaus

I don't understand how the 1-2 pair in Fig.2 A-6 and G-6 can remove 1-2 in B5, can you please elaborate on this.Thank You,Klaus

BTY you have a great site, its so informative and I have learned so much from it, greatly appreciated.

Andrew Stuart writes:

Candidates 1/2 are the *only* candidates in cells A6 and G6. You notice that B6 has an extra candidate 8. If all three only had 1/2 we'd be in trouble, we'd have gone wrong on some previous step. 1/2 *must* go in A6 and G6 so it leaves no room for 1/2 eslewhere in the columns. Fortunately that 8 allows us to fill B6 and move on.

Friday 26-Apr-2013

... by: Roy

Just realized there is another Naked Quad in Box 1 in the Naked Quad example (2,3,4,7) in cells A2, A3, B3 & C3! Alas, it doesn't really help reduce any other cells. With two Naked Quads in the Box, the only other number is 9 which has already been identified in cell C2.

Friday 5-Apr-2013

... by: Roy

The logic of cell B2 being 1 or 8 for the Naked Quads section is derived from the fact one of the cells A1, B1 & C1 will contain either a 1 or 8 but not both as cell H1 will have the other. Cell B2 is the only other cell in this block that has either of these two numbers. I guess this would be a hidden pair within a pair of naked quads.

Monday 25-Mar-2013

... by: Roy

Having reduced cell B2 to possibles 1 or 8, cell B1 can be reduced to possibles 5 or 6 (hidden pair with cell B4) and then H1 must be 8 as would be the case for B2! A1, B1 & C1 have been reduced to a naked triple (1,5,6) with a {2,2,2} formation!

Friday 22-Mar-2013

... by: Roy

I believe you can go one step further with cell B2 in the Naked Quads example. It will have a value of either 1 or 8 given the restrictions from cell H1.

Monday 21-Jan-2013

... by: Anon

i am really thankful to you for presenting this.really very useful content about sudoku solving. i am impressed.

Saturday 10-Nov-2012

... by: Peter Rogers

HiWhat a fabulous website. This is what I have been looking for for years a real how to solve sudokus. I am stunned at its teaching capacity.

Wednesday 5-Oct-2011

... by: gerhard, sweden

Assume that a triplet consists of the three bigrams(example) 56, 67, 57, occuring in one row.

Let´s say that the bigrams occur in region 1 (bigrams 56 and 67) and in another region (57).

It is obvious that they work the same as any true triplet, but less obvious that the figure 6 can be eliminated from the remaining squares of region 1.

If this is described somewhere else, please excuse me for commenting.

Tuesday 6-Sep-2011

... by: Pieter, Newtown, Australia

Hi AndrewI always love to double-check my solution to a puzzle using your solver. I got this one by XY-Chains but damn it! I missed the naked & hidden Quads, yet again! I usually do, damn quads! :-(

I noticed you don't have an example for quads in your "Pick an Example" drop-down list. Want to include this one?

It's from the Sydney Sun-Herald of 2011-09-4 (Auspac Media for the puzzle). You may need to check with them re copyright.

Thanks as always for your great solver!Ciao, Pieter

Andrew Stuart writes:

Excellent example, thank you for sharing. I don't think I can use it in my example list but it can be linked here as you have stated the credits.

Wednesday 22-Jun-2011

... by: Charlie R

Wow! What a site. I landed here by chance. I have been exploring sudoku myself, using my own excel-based solver, convinced that there must be a complete rule-based solution. I had found many of the rules myself, but this is a much more complete set, beautifully explained and illustrated. I bow to you, Oh Master

Monday 4-Apr-2011

... by: Dayanandan

Landed on this site by chance. The joy experienced is such that I want to tell you this at once. This clears my doubt fully. I like this illustration as well.

RegardsDayanandan

Monday 7-Mar-2011

... by: hutch

i thought i could write this in excel and i did get some parts but soon realized the complexity and have stopped(at least 4 now). the stepwise debugger style is the bomb. i hope to improve my sudoku but i think i will spend a good bit of time just admiring this work.many thanks for the obvious labor of love.hutchpawleys

Saturday 5-Mar-2011

... by: Andrew

Almost a year later, a response to Mike, who said:

"I notice that Sudoku Solver does not exhaustively identify all naked pairs as seen in the following puzzle.

In row E the 2,5 pair in columns 3 and 7 should reduce cell E9 to just 1."

I don't understand why you say that. There's not such a naked pair there, and E9 has already a 3... Perhaps the sudoku saved with that id changed?

Tuesday 15-Feb-2011

... by: Blaster88

Non seulement c'est génial mais en plus je bosse mon anglais !

Tuesday 15-Jun-2010

... by: John

After reading this over, I think I understand why naked triples (and naked quadruples and quintuples). If you understand how naked pairs work, look at naked triples this way: When you solve one of the 3 cells, the other two cells become naked pairs or single. Then all three numbers in a naked triple can be eliminated from the other cells.

For example: (123) (123) (123)Make any of the cells a 1: (123) (1) (123)Drop 1 from the other cells: (23) (1) (23) You can delete 1 from all other cells, because it is used. You can eliminate 23 from the other cells because it is a naked pair.

The same works for other triples: (123) (12) (23)If the middle cell is 1: (23) (1) (23)Eliminate 23 from other cells because it is a naked pair. and so forth...

Thursday 27-May-2010

... by: Michael

To all who are having difficulty understanding this...

A naked pair shows the same two values and only those values in two different fields (in the same column, row, or three by three square). This shows that those two fields each must have one of the two values (there are no other values to choose from). Since a value cannot occur more than once in any one column, row, or three by three square) the two values can be safely removed from the other clues since it is know that they must appear in the place of the naked pair.

Naked Triples and Quads simply extend the same logic to 3 and 4 values.

Wednesday 31-Mar-2010

... by: Mike

I notice that Sudoku Solver does not exhaustively identify all naked pairs as seen in the following puzzle.

In row E the 2,5 pair in columns 3 and 7 should reduce cell E9 to just 1.

Andrew Stuart writes:

The solver is working correctly but the behavior in these cases is worth explaining. As Naked Pairs are detected the removal effects are applied. This might occasionally stop another Naked Pair being found since some numbers have be removed. The solver *could* detect all NPs and then apply the results simultaneously but for speed and space I have chosen not to. Usually the next set of NPs will be discovered in the next round. This applies to most of the basic strategies.

Friday 12-Mar-2010

... by: Pete

I've been looking for help and this is the first I've seen that looks like it will help. Bring on the 6 star puzzles. I'm ready(I think).

Thursday 25-Feb-2010

... by: CS VIDYASAGAR

Excellent explanation with very useful examples to make one understand difficult concepts naked pairs and naked triples.Thanks for keeping the aritcle simple and easily understandable.

Tuesday 5-Jan-2010

... by: Harpo

I agree with buc; with the information given it still seems rather illogical to remove the other candidates.

Tuesday 8-Dec-2009

... by: Werty

My explanation of naked triples.On the example.imagine that you put 5 in one of the columns 2, 3 or 4. That will leave only 7 and 8 as candidates in three columns - 1, 8 and 9. Clear?You will get to similar wrong position when you put 8 in column 4.

Tuesday 8-Dec-2009

... by: Carol Kennedy

I am just learning this game and so enjoy it. But I do not always understand your lessons. For example, if you have 4,84,8 in a row then you can eliminate the other 4,8s in that row, but can I also erase all the other 4,8s in the column and the entire box as well? Thank you.

Sunday 16-Aug-2009

... by: Curt Klemenz

I'm in same boat...having ultimate difficulty spotting hidden pairs and triples. When they are pointed out, .... I see them.

I suspect there is a mental algorithm for focusing attention toward the specific candidates, but no luck so far.

Anyone with a suggestion that's willing to share?

Monday 3-Aug-2009

... by: Bruce D

An explanation on how the naked tripple works. As in the example shown, we have (7,8) (5,7,8) and (5,7,8). The first cell can contain a 7 or an 8. That means that the other two cells will then contain a 5 and 8 in the case the first one is a 7, or a 5 and 7 if the first cell is an 8. By having the last two cells being conditional on the other, we can eliminate the 5, 7, 8 from all other cells in the row.

Saturday 6-Jun-2009

... by: Rockmelon

I have been an accountant for 35 years (which means nothing) and I can't see the relationships among these numbers! I have a really difficuolt time understanding this and I love to do Sudoku!

Any suggestions??

Sunday 10-May-2009

... by: BobCarl

As you know, any row, column or box contains nine cells.

When there are only 3 different numbers that can fit into three of the nine cells, that automatically eliminates their use in the remaining six cells. Hence, they can be removed as candidates from those "other cells".

Monday 20-Apr-2009

... by: buc

Re naked tripple: I would appreciate you explaining the logic of removing any of the three candidates from other cells.

Article created on 9-June-2005. Views: 506441This page was last modified on 31-January-2012.All text is copyright and for personal use only but may be reproduced with the permission of the author.
Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2012